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Reley-Jins qonuni

Reley-Jins qonuni

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Reley qonuni
Tarixiy rivojlanishPlank qonuni bilan taqqoslashChastota va to'lqin uzunligiga bog'liq ifodalarning izchilligiReley-Jins qonunining boshqa shakllariYana qarangFoydalanilgan adabiyotlar ro'yxatiHavolalar

Fizikada Reley-Jins qonuni klassik dalillar orqali ma'lum bir haroratdagi qora jismning to'lqin uzunligiga funksiyasi sifatida elektromagnit nurlanishning spektral nurlanishiga yaqinlikdir. To'lqin uzunligi λ uchun bu: B λ ( T ) = 2 c k B T λ 4 {\displaystyle B_{\lambda }(T)={\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}} {\displaystyle B_{\lambda }(T)={\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}}bu yerda B λ {\displaystyle B_{\lambda }} {\displaystyle B_{\lambda }} - spektral nurlanish, birlik chiqarish maydoniga, steradianga, to'lqin uzunligi birligiga chiqariladigan quvvat; c {\displaystyle c} {\displaystyle c} yorug'lik tezligi ; k B {\displaystyle k_{\mathrm {B} }} {\displaystyle k_{\mathrm {B} }} Boltsman doimiysi ; va T {\displaystyle T} {\displaystyle T} Kelvinda o'lchanadigan harorat . Chastota ν {\displaystyle \nu } {\displaystyle \nu } uchun, yuqoridagi ifoda o'rniga B ν ( T ) = 2 ν 2 k B T c 2 {\displaystyle B_{\nu }(T)={\frac {2\nu ^{2}k_{\mathrm {B} }T}{c^{2}}}} {\displaystyle B_{\nu }(T)={\frac {2\nu ^{2}k_{\mathrm {B} }T}{c^{2}}}}Reley-Jins qonuni katta to'lqin uzunliklarida (past chastotalar) eksperimental natijalarga mos keladi, ammo qisqa to'lqin uzunliklarida (yuqori chastotalar) katta farq mavjud. Kuzatishlar va klassik fizikaning bashoratlari o'rtasidagi bu nomuvofiqlik ultrabinafsha falokat sifatida tanilgan.[1][2] Barcha chastotalarda to'g'ri nurlanishni ta'minlaydigan Plank qonuni past chastotalar chegarasi sifatida Reley-Jins qonunini kabidir.

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Harorati 5800K jism uchun Reyl-Jins qonunini Vin yaqinlashuvi va Plank qonuni bilan solishtirish.

1900 yilda ingliz fizigi Lord Reley teng taqsimot teoremasiga tayangan holda, klassik fizik dalillarga asoslanib Relley-Jins qonunining λ − 4 {\displaystyle \lambda ^{-4}} {\displaystyle \lambda ^{-4}}ga bog'liqligini chiqardi. Ushbu qonun to'lqin uzunligi nolga yaqinlashganda (chastota cheksizlikka intilganligi sababli) cheksizlik tomon ajraladigan energiya chiqishini bashorat qildi. Haqiqiy qora jismlarning spektral emissiyasini o'lchash shuni ko'rsatdiki, emissiya past chastotalarda Reley hisobiga mos keladi, lekin yuqori chastotalarda farqlanadi; maksimal darajaga yetadi va keyin chastota bilan tushadi, shuning uchun chiqarilgan umumiy energiya cheklangan. Reley o'z formulasining yuqori chastotalarda fizik bo'lmagan xatti-harakatlarini tan oldi va uni tuzatish uchun maxsus kesimni kiritdi, ammo eksperimentatorlar uning kesimi ma'lumotlarga mos kelmasligini aniqladilar.[1][3] Xendrik Lorens ham1903 yilda to'lqin uzunligiga bog'liqligining xulosasini taqdim etdi. Proporsionallik konstantasini o'z ichiga olgan to'liqroq xulosalsr 1905 yilda Reley va Ser Jeyms Jins va mustaqil ravishda Albert Eynshteyn tomonidan taqdim etilgan.[3] Rayleigh bu tafovutni yuqori chastotali tebranishlar uchun haqiqiy bo'lmagan teng taqsimot teoremasi orqali hal qilish mumkin deb hisoblardi, Jeans esa asosiy sabab materiya va yorug'lik efirining issiqlik muvozanatida bo'lmasligini ta'kidladi.[3]

Reley chastotaga bog'liqlik haqidagi birinchi xulosasini 1900 yil iyun oyida nashr etdi. Plank o'sha yilning oktyabr oyida hozirda Plank qonuni deb nomlanuvchi egri chiziqni kashf etdi va dekabrda taqdim etdi.[3] Plankning asl maqsadi yuqori chastotalarda ma'lumotlarni aniq tasvirlaydigan absolut qora jism nurlanish egri chizig'i uchun Vin ifodasining qoniqtiruvchi xulosasini topish edi. Plank Vinning dastlabki xulosalarini noadekvat deb topdi va o'zining formulalarini ishlab chiqdi. Keyin, so'nggi eksperimental natijalar uning past chastotalar bo'yicha bashoratlari bilan mos kelmasligini bilib, Plank o'z xulosalarini qayta ko'rib chiqdi va hozir Plank qonuni deb ataladigan qonunga erishdi.[4]

1900 yilda Maks Plank qora jism nurlanishining to'lqin uzunligi λ = c/ν ( Plank qonuni ) bilan ifodalangan ifodasini empirik ravishda oldi: B λ ( T ) = 2 h c 2 λ 5   1 e h c λ k B T − 1 {\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}} {\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}}bu yerda h {\displaystyle h} {\displaystyle h} - Plank doimiysi va k B {\displaystyle k_{B}} {\displaystyle k_{B}} - Boltsman doimiysi . Plank qonuni ultrabinafsha falokatidan aziyat chekmaydi va eksperimental ma'lumotlarga yaxshi mos keladi, lekin uning to'liq ahamiyati (oxir-oqibat kvant nazariyasiga olib keldi) bir necha yil o'tgach baholandi. Chunki, e x = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ {\displaystyle e^{x}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+\cdots } {\displaystyle e^{x}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+\cdots }keyin yuqori haroratlar yoki uzun to'lqin uzunliklari chegarasida eksponensialdagi had kichik bo'ladi va eksponensial Teylor polinomining birinchi tartibli hadi bilan yaqinlashadi, e h c λ k B T ≈ 1 + h c λ k B T {\displaystyle e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}\approx 1+{\frac {hc}{\lambda k_{\mathrm {B} }T}}} {\displaystyle e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}\approx 1+{\frac {hc}{\lambda k_{\mathrm {B} }T}}}Shunday qilib, 1 e h c λ k B T − 1 ≈ 1 h c λ k B T = λ k B T h c {\displaystyle {\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {1}{\frac {hc}{\lambda k_{\mathrm {B} }T}}}={\frac {\lambda k_{\mathrm {B} }T}{hc}}} {\displaystyle {\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {1}{\frac {hc}{\lambda k_{\mathrm {B} }T}}}={\frac {\lambda k_{\mathrm {B} }T}{hc}}}Buning natijasida Plankning qora jism formulasi quyidagiga qisqaradi B λ ( T ) = 2 c k B T λ 4 {\displaystyle B_{\lambda }(T)={\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}} {\displaystyle B_{\lambda }(T)={\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}}bu ifoda klassik Reley-Jins ifodasi bilan bir xil.

Xuddi shu argumentni ν = c λ {\displaystyle \nu ={\frac {c}{\lambda }}} {\displaystyle \nu ={\frac {c}{\lambda }}} chastota ifodasi bilan ifodalangan qora jism nurlanishga nisbatan qo'llanilishi mumkin. Kichik chastotalar chegarasida, ya'ni h ν ≪ k B T {\displaystyle h\nu \ll k_{\mathrm {B} }T} {\displaystyle h\nu \ll k_{\mathrm {B} }T} , B ν ( T ) = 2 h ν 3 c 2 1 e h ν k B T − 1 ≈ 2 h ν 3 c 2 ⋅ k B T h ν = 2 ν 2 k B T c 2 {\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {2h\nu ^{3}}{c^{2}}}\cdot {\frac {k_{\mathrm {B} }T}{h\nu }}={\frac {2\nu ^{2}k_{\mathrm {B} }T}{c^{2}}}} {\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {2h\nu ^{3}}{c^{2}}}\cdot {\frac {k_{\mathrm {B} }T}{h\nu }}={\frac {2\nu ^{2}k_{\mathrm {B} }T}{c^{2}}}}Bu oxirgi ifoda kichik chastotalar chegarasidagi Reley-Jins qonunidir.

Reley-Jins qonunining chastota va to'lqin uzunligiga bog'liq ifodalarini solishtirganda shuni yodda tutish kerakki, d P d λ = B λ ( T ) {\displaystyle {\frac {dP}{d\lambda }}=B_{\lambda }(T)} {\displaystyle {\frac {dP}{d\lambda }}=B_{\lambda }(T)}va d P d ν = B ν ( T ) {\displaystyle {\frac {dP}{d\nu }}=B_{\nu }(T)} {\displaystyle {\frac {dP}{d\nu }}=B_{\nu }(T)}Shuning uchun, B λ ( T ) ≠ B ν ( T ) {\displaystyle B_{\lambda }(T)\neq B_{\nu }(T)} {\displaystyle B_{\lambda }(T)\neq B_{\nu }(T)}qiymatni almashtirgandan keyin ham λ = c / ν {\displaystyle \lambda =c/\nu } {\displaystyle \lambda =c/\nu }, chunki B λ ( T ) {\displaystyle B_{\lambda }(T)} {\displaystyle B_{\lambda }(T)} vaqt birligi ichida yuzaning birlik maydoniga chiqariladigan energiya birliklariga, qattiq burchak birligiga, to'lqin uzunligi birligiga ega, shu bilan birga B ν ( T ) {\displaystyle B_{\nu }(T)} {\displaystyle B_{\nu }(T)} aqt birligi ichida yuzaning birlik maydoniga chiqariladigan energiya birliklariga, qattiq burchak birligiga, chastota birligiga ega. Barqaror bo'lish uchun biz quyidagi tenglikdan foydalanishimiz kerak B λ d λ = d P = B ν d ν {\displaystyle B_{\lambda }\,d\lambda =dP=B_{\nu }\,d\nu } {\displaystyle B_{\lambda }\,d\lambda =dP=B_{\nu }\,d\nu }bu yerda endi ikkala tomonda ham chiqarish yuzasining birlik maydoniga, birlik qattiq burchakka quvvat birliklari (vaqt birligi uchun chiqarilgan energiya) mavjud.

To'lqin uzunligi bo'yicha Reyl-Jins qonunidan boshlab biz quyidagini olamiz B λ ( T ) = B ν ( T ) d ν d λ {\displaystyle B_{\lambda }(T)=B_{\nu }(T)\,{\frac {d\nu }{d\lambda }}} {\displaystyle B_{\lambda }(T)=B_{\nu }(T)\,{\frac {d\nu }{d\lambda }}}bu yerda d ν d λ = d d λ ( c λ ) = − c λ 2 {\displaystyle {\frac {d\nu }{d\lambda }}={\frac {d}{d\lambda }}\left({\frac {c}{\lambda }}\right)=-{\frac {c}{\lambda ^{2}}}} {\displaystyle {\frac {d\nu }{d\lambda }}={\frac {d}{d\lambda }}\left({\frac {c}{\lambda }}\right)=-{\frac {c}{\lambda ^{2}}}}Bizga quyidagini topishsga undaydi: B λ ( T ) = 2 k B T ( c λ ) 2 c 2 × c λ 2 = 2 c k B T λ 4 {\displaystyle B_{\lambda }(T)={\frac {2k_{\mathrm {B} }T\left({\frac {c}{\lambda }}\right)^{2}}{c^{2}}}\times {\frac {c}{\lambda ^{2}}}={\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}} {\displaystyle B_{\lambda }(T)={\frac {2k_{\mathrm {B} }T\left({\frac {c}{\lambda }}\right)^{2}}{c^{2}}}\times {\frac {c}{\lambda ^{2}}}={\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}}

Qo'llanilishiga qarab Plank funksiyasi 3 xil ko'rinishda ifodalanishi mumkin. Birinchisi, nurlanish yuzasining birlik maydoniga, birlik qattiq burchakka, spektr birligiga vaqt birligida chiqariladigan energiyani o'z ichiga oladi. Ushbu ifoda Plank funksiyasi va unga bog'liq bo'lgan Rley-Jins chegaralari bilan berilgan B λ ( T ) = 2 h c 2 λ 5   1 e h c λ k B T − 1 ≈ 2 c k B T λ 4 {\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}} {\displaystyle B_{\lambda }(T)={\frac {2hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}}}yoki B ν ( T ) = 2 h ν 3 c 2 1 e h ν k B T − 1 ≈ 2 k B T ν 2 c 2 {\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {2k_{\mathrm {B} }T\nu ^{2}}{c^{2}}}} {\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {2k_{\mathrm {B} }T\nu ^{2}}{c^{2}}}}Shu bilan birga, barcha qattiq burchaklar bo'ylab integratsiyalangan emission quvvat uchun Plank qonunini o'rnida ushbu ifoda I ( ν , T ) = π B ν ( T ) {\displaystyle I(\nu ,T)=\pi B_{\nu }(T)} {\displaystyle I(\nu ,T)=\pi B_{\nu }(T)} ko'rinishida yozish mumkin. Ushbu shaklda Plank funksiyasi va unga bog'liq bo'lgan Reley-Jins chegaralari bilan berilgan I ( λ , T ) = 2 π h c 2 λ 5   1 e h c λ k B T − 1 ≈ 2 π c k B T λ 4 {\displaystyle I(\lambda ,T)={\frac {2\pi hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {2\pi ck_{\mathrm {B} }T}{\lambda ^{4}}}} {\displaystyle I(\lambda ,T)={\frac {2\pi hc^{2}}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {2\pi ck_{\mathrm {B} }T}{\lambda ^{4}}}}yoki I ( ν , T ) = 2 π h ν 3 c 2 1 e h ν k B T − 1 ≈ 2 π k B T ν 2 c 2 {\displaystyle I(\nu ,T)={\frac {2\pi h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {2\pi k_{\mathrm {B} }T\nu ^{2}}{c^{2}}}} {\displaystyle I(\nu ,T)={\frac {2\pi h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {2\pi k_{\mathrm {B} }T\nu ^{2}}{c^{2}}}}Boshqa hollarda Plank qonuni birlik hajmdagi energiya uchun (energiya zichligi) ushbu ko'rinishfa u ( ν , T ) = 4 π c B ν ( T ) {\textstyle u(\nu ,T)={\frac {4\pi }{c}}B_{\nu }(T)} {\textstyle u(\nu ,T)={\frac {4\pi }{c}}B_{\nu }(T)} yoziladi. Ushbu shaklda Plank funksiyasi va unga bog'liq bo'lgan Reley-Jins chegaralari bilan berilgan u ( λ , T ) = 8 π h c λ 5   1 e h c λ k B T − 1 ≈ 8 π k B T λ 4 {\displaystyle u(\lambda ,T)={\frac {8\pi hc}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {8\pi k_{\mathrm {B} }T}{\lambda ^{4}}}} {\displaystyle u(\lambda ,T)={\frac {8\pi hc}{\lambda ^{5}}}~{\frac {1}{e^{\frac {hc}{\lambda k_{\mathrm {B} }T}}-1}}\approx {\frac {8\pi k_{\mathrm {B} }T}{\lambda ^{4}}}}yoki u ( ν , T ) = 8 π h ν 3 c 3 1 e h ν k B T − 1 ≈ 8 π k B T ν 2 c 3 {\displaystyle u(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {8\pi k_{\mathrm {B} }T\nu ^{2}}{c^{3}}}} {\displaystyle u(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}\approx {\frac {8\pi k_{\mathrm {B} }T\nu ^{2}}{c^{3}}}}

  • Stefan-Bolsman qonuni
  • Vinning siljish qonuni
  • Sakuma-Xattori tenglamasi
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    Kutner, Mark L.. Astronomy: A Physical Perspective. Cambridge University Press, 2003 — 15-bet. ISBN 0-521-52927-1.
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    Rybicki. Radiative Processes in Astrophysics. Wiley, 2004 — 20–28-bet. ISBN 0-471-82759-2.
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    Pais, A. (1979-10-01). „Einstein and the quantum theory“. Reviews of Modern Physics (inglizcha). 51-jild, № 4. 863–914-bet. doi:10.1103/RevModPhys.51.863. ISSN 0034-6861.
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    Kragh, H. (2000). „Max Planck: the reluctant revolutionary“. Physics World. 13-jild, № 12. 31–36-bet. doi:10.1088/2058-7058/13/12/34. ISSN 0953-8585.

Ushbu tarjima O'zbekiston Milliy Universiteti Fizika fakulteti talabasi Ikbalova Muxlisaxon tomonidan qilindi.

  • Reley-Jins qonunining kelib chiqishi
  • Kubdagi to'lqin rejimlarini hosil qilish
Bu maqola birorta turkumga qoʻshilmagan. Iltimos, maqolaga aloqador turkumlar qoʻshib yordam qiling. (Aprel 2024)
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Tarixiy rivojlanish

Plank qonuni bilan taqqoslash

Chastota va to'lqin uzunligiga bog'liq ifodalarning izchilligi

Reley-Jins qonunining boshqa shakllari

Yana qarang

Foydalanilgan adabiyotlar ro'yxati

Havolalar